Square root calculator employing a modified sum of the odd integers method



Sept. 1, 1970 R. A. RAGEN 3,526,750

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United States Patent Office 3,526,760 Patented Sept. 1, 1970 SQUARE ROOT CALCULATOR EMPLOYING A MODIFIED SUM OF THE ODD INTEGERS METHOD Robert A. Ragen, Hayward, Calif., assignor, by mesne assignments, to The Singer Company, a corporation of New Jersey Filed Apr. 1, 1966, Ser. No. 539,569 Int. Cl. G06f 7/48 US. Cl. 235-158 18 Claims ABSTRACT OF THE DISCLOSURE A calculator is disclosed for deriving the square root of an operand according to a modified sum of the odd integers method. The square root is obtained by arranging the operand in consecutive pairs of contiguous digits and performing a plurality of series of successive substractions on several partial operands. Each partial operand comprises a pair of digits arranged to the right of the last non-negative remainder from the previous series of subtractions. Circuitry is provided for decimally aligning the entered operand and the derived square root in accordance with the number of digits desired to the right of the decimal point.

This invention relates to improvements in electronic calculators and more particularly, but not by way of limitation, to an electronic calculator which simply and quickly derives the square root of a number, the square root being correctly decimally aligned. The invention also provides for the cumulative counting of the entry of each complete number into the calculator.

Electronic computers with the capability of extracting the square root of an operand are known in the art, and have provided a significant improvement in the operation of numbers over mechanical calculating machines. There are digital computers, presently known in the art, which provide highly accurate and extremely fast calculations. By utilizing sophisticated circuitry, these present digital computers can be adapted to derive the square root of a number. However, due to the extraordinary speed requirements of these machines, this added capability results in a disproportionate increase in cost.

In the presently emerging market for deck-top digital computers and electronic calculating machines, simplicity of design and reasonableness of cost are important considerations. The cost of adapting a desk-top electronic calculator to derive a square root usingpresent known methods would be economically prohibitive. Also, While the speed requirements of a deck-top electronic calculator are lower than that of a large digital computer, nonetheless, speed is a consideration. It is desirable in the derivation of a correct decimally aligned square root by a desk-top electronic calculator that the time required for this derivation not to differ substantially from that required for the performance of other functions ofthe machine such as multiplication and division.

It is also desirable for an operator of the electronic calculator to be able to easily perform a square root operation with only an absolute minimum manipulation of the keyboard. An advantage of electronic calculators over rotary mechanical calculating machines is that many arithmetic functions may be performed in a simpler manner, thereby permitting use of the machine by a relatively untrained operator. In furtherance of the objective of permitting complete operation of the calculator by an operator possessing only limited training, the square root function of the calculator must be initiated with substantially the same case as other arithmetic functions performed by the calculator.

When a series of numbers are entered into the calculator by manipulation of the keyboard, it is considered desirable that a cumulative sum of the entries be temporarily recorded so that a running check of the computation in a problem may be maintained. This is of considerable assistance, not only to casual users of the calculator with limited experience, but also to skilled operators of calculating machines.

The novel method employed in the present invention for deriving a square root is a modified sum of the odd integers method. The sum of the odd integers method is known as a method of extracting a square root, but it is not easily adapted to digital techniques, and until the present invention, a practical manner of utilizing this method had not been known.

Odd integers are the whole numbers 1, 3, 5, 7, 9, 11, etc. If such a succession of odd integers (an odd order arithmetic progression) are added, each summation will result in a perfect square. A perfect square is, of course, a number whose square root is a whole number. This is illustrated by the following table:

By adding the next higher odd order integer to each preceding summation, all existing perfect squares are produced, with none being omitted. It should also be noted that the number of odd order integers in each summation is the square root of the sum. In example five of the preceding table, there are five parts to the summation, and five is the square root of twenty-five. This observation is true for any combination of successive odd integers.

The square root of a number may be extracted by re versing this procedure. In this instance, the successive odd integers starting at one are subtracted to arrive at the square root. The following example illustrates this:

The number of successive subtractions is the square root.

This is a satisfactory method for use with perfect squares. A related method of deriving the square root is accomplished by applying a simple formula rather than counting the subtractions to arrive at the square root. This formula is represented by the equation R=.(N+1)/2 where N is the last integer to be successfully subtracted (no overdraft), which is sometimes called the partial root and R represents the square root.

In an example of extracting the square root of 64, it is obvious that the last number that could be successfully subtracted would be fifteen. By adding one to fifteen, which would be sixteen, and then dividing by two, the eight would be the square root of sixty-four.

This method, which has been presently shown, is valid for small numbers but is unduly cumbersome for large numbers. Since the number of subtract cycles is equal to the square root of the number, an excessive length of time is necessary to complete a problem using large num- 

